Definition 2 (Prime Order Bilinear Group [1]}) 
Let \(\mathbb {G}_0\)  and \(\mathbb {G}_1\)  be two multiplicative cyclic groups of same prime order p. g is an arbitrary generator \(g\xleftarrow{}\mathbb {G}_0\) . e is a symmetric bilinear map, such that \(e:\mathbb {G}_{0} \times \mathbb {G}_{0}\rightarrow \mathbb {G}_{1}\)  where \(e\left( g^{x} ,g^{y}\right) =e\left( g^{y} ,g^{x}\right) =e( g,g)^{xy} =e( g,g)^{yx}\) .

